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Theorem dyadval 23560
Description: Value of the dyadic rational function 𝐹. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
dyadmbl.1 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
Assertion
Ref Expression
dyadval ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dyadval
StepHypRef Expression
1 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
2 oveq2 6821 . . . 4 (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵))
31, 2oveqan12d 6832 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵)))
4 oveq1 6820 . . . 4 (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
54, 2oveqan12d 6832 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵)))
63, 5opeq12d 4561 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
7 dyadmbl.1 . 2 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
8 opex 5081 . 2 ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩ ∈ V
96, 7, 8ovmpt2a 6956 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cop 4327  (class class class)co 6813  cmpt2 6815  1c1 10129   + caddc 10131   / cdiv 10876  2c2 11262  0cn0 11484  cz 11569  cexp 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818
This theorem is referenced by:  dyadovol  23561  dyadss  23562  dyaddisjlem  23563  dyadmaxlem  23565  opnmbllem  23569  opnmbllem0  33758
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